SANDIA REPORT SAND81-1618· Unlimited Release· UC-62 Printed November 1981

A Game Theory Approach to Consumer Incentives for Solar Energy

John K. Sharp

Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 for the United States Department of Energy under Contract DE-AC04-76DPOO789

SF2900Q(S-S1 )

When printing a copy of any digitized SAND Report, you are required to update the

markings to current standards.

Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Govern­ment nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or pro­cess disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof or any of their contractors or subcontractors.

Printed in the United States of America Available from National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road Springfield, VA 22161

NTIS price co4lls... Printed copy: :;; ':1 • 0 0 Microfiche copy: A01

SANDBl-161B Unlimited Release

Printed November 19B1 Distribution

Category uc - 60

A GAME THEORY APPROACH TO CONSUMER INCENTIVES FOR SOLAR ENERGY

John K. Sharp Photovoltaic Systems Development Division 4723

Sandia National Laboratories Albuquerque, New Mexico

ABSTRACT

Solar energy is currently not competitive with fossil fuels . Fossil fuel price increases may eventuallY allow solar to compete, but in­centives can change the relative price between fossil fuel and solar energy, and make solar compete sooner. This paper develops examples of a new type of competitive game using solar energy incentives. Compet ­itive games must have players with individual controls and conflicting objectives, but recent work also includes incentives offered by one of the players to the others. In the incentive game presented here, the Government acts as the leader and offers incentives to consumers, who act as followers. The Government incentives · offered in this leader­follower (Stackelberg) game reduce the cost of solar energy to the consumer. Both the Government and consumers define their own objec­tives with the Government determining an incentive (either in the form of a subsidy or tax) that satisfies its objective. The two hypotheti ­cal examples developed show how the Government can achieve a stated solar utilization rate with the proper incentives. In the first example the consumer's utility function guarantees some purchases of solar energy. In the second example~ the consumer's utility function allows for no solar purchases because utility is derived only from the amount of energy used and not from the source of the energy. The two examples discuss both sUbsidy ~nd tax incentives, with the best control over solar use coming from fossil fuel taxes dependent upon the amount of solar energy used. Future work will expand this static analysis to develop time varying incentives along a time and quantity dependent learning curve for the solar lndustry.

This work was supported by the U. S. Department of Energy

3

4

A GAME THEORY APPROACH TO CONSUMER INCENTIVES FOR SOLAR ENERGY

Introduction

Solar energy incentives can improve the relative economics between

solar and fossil fuel. Presently, solar energy is not economical when

compared to fossil fuel, but with future fossil fuel price increases

and solar energy price decreases, solar may become competitive in

certain locations. Government incentives can help make solar energy

competitive sooner by stimulating and developing the solar industry.,

The result~ng solar production increases will lower system costs.

Determining the proper incentive mi~ for the Government to offer is a

major problem because consumers' and suppliers' reactions to the in­

centives must be considered. The economic problems associated with

initial overstimulation and/or discontinuous stimulation can be avoided

with the proper incentive prog~am. The Government acting as a leader

can offer incentives which encourage consumers to purchase more solar

energy. By calling the Government a leader and the consumers followers ,

a generalized problem can be developed where the leader has his own

objectives, and he is trying to influence purchases of the followers

who have their own objectives. In the literature, this type of leader­

follower game is called a Stackelberg game l Solar incentives will now

be analyzed using Stackelberg game theory.

2 Game theory is the study of players who can a6t upon and change a

system. Players attempt to make favorable changes in their individual

cos t or objective functions. Games are either cooperative or competi­

tive with the players acting accordingly. Stackelberg games 3,4 are

competitive and contain two types of players: a leader and followers.

The leader in a Stackelberg game knows how the followers react to him,

and he uses this information to choose his most desirable action. Game

theory has recently been used to investigate incentives in public

decisionmaking 5 ; incentives have also been used with Stackelberg games

to investigat e duoPolies. 6

Solar Incentives Using Stackelberg Games

Incentive theory for solar energy can be developed using Stackel­

berg games. The ultimate goal of an incentive program is to make solar

energy competitive with other energy forms. If this goa l is not

achieved, then incentives must be continued after the solar industry

has matured. This time varying problem can be treated as static if

intermediate targets are set for the solar industry. The Government

acting as a leader has a specific short-term target it hopes to achieve

with incentives while consumers acting as followers have their own

objectives. The Government's short-term objective could be a produc­

tion target while the consumers' opjectives are to maximize their well­

being (commonly referred to as utility in economics) subject to the

funds they have available with which to purchase energy. Using the

single consumer case, these conditions can be expressed analytically as:

Leader (Government) Objective: achieve a stated goal J

Follower (Consumer) Objective: maximize utility U

where:

Subject to:

ql = units of fossil energy

q2 units of solar energy

Pl = price per unit of fossil fuel

P2 = price per unit of solar fuel

I total consumer budget for energy products

(3 )

The model described by (1) ~ (3) does not include Government

incentives. In this model · the consumer maximizes his utility subject to

the budget constraint to determine his purchases ql* and q2* (with q*

denoting the optimal purchases). These values determine the deviation

from the Government's objective. The three types of Government in­

centives available to change the consumers' energy mix are:

1. Solar energy subsidy

2. Fossil energy tax

3. Rationing and/or regulating energy purchases

A solar energy subsidy allows the consumer to purchase more solar

energy with the same amount of funds. Fossil fuel taxes decrease the

amount of fossil fuel that can be purchased and encourage the use of

5

6

solar energy by reducing the relative price between solar and fossil

energy. Rationing fossil energy forces consumers to switch to solar

for additional energy, and regulating or requiring more solar usage

increases the demand for solar. The budget constraint (3) modified to

account for subsidy and tax incentives is:

(4)

where:

t per unit fossil fuel tax

s = per unit solar

Ratioring or regulating is where both sand t are zero, and the

Government determines the value of ql or q2.

The subsidy or tax must ensure that, when the consumer acts in

his own best interest, the Government's objective is realized. The

Government can achieve this by first characterizing the consumers'

reaction to various taxes and/or subsidies. This information is then

used as a constraint when the Government tries to achieve its object­

ive. Two hypothetical examples are presented which show how the

Government can determine the necessary incentives to achieve a targeted

production level of solar energy.

Stacke lb erg Example 1

The two examples differ only in the utility derived by the consumer

from energy. In both examples the Government's objective is to achieve

a targeted solar production level. This production level lsassumed to

be a point along a learning curve for the solar industry. The first

example assumes a standard consumer utility function which is the fossil

fuel usage times the solar energy usage.

The competitive game occurs when the consumer tries to maximize

his utility subject to his budget constraintwhlle th~ Government tries

to ach i eve the targeted level of solar energy usage. The objective

functions and budget constraint for this example are:

Leader's (Government's) Goal: minimize the target solar pro­duction error

J 2

(q2 - 0.4) (5)

~ -~ D 0::8 ~ .... z ~

0:: <~ ~cS 0 Ul

~o 01C1 00 0

E-t ....... Z ~~ II cS

C\J

~8 d

--.J

, • • • • • , , , , , •

0.0

· • • · • \

, ,

. ,

, •

Figure 1.

., , , , ,

\ , , , \ ,

, \

, , ., , , , ,

, \ ,

\

,

, , , ,

, ,

, ,

, , , , , , ,

t=-7·5qz+3.

, , , ,

~~-

" "

-~-

~~~

- ... --

----------------

1.0 2.0 3.0 4.0

FUEL 5.0 6.0

ql UNI TS OF FOSS I L

Impact of Incentives on The Consumer ' s Energy Use Assuming a Stanaard Utility Function

8

Followers' (Consumers') Goal: maximize utility

(6 )

Subject to the budget contraint:

(7)

Solar and fossil energy have the same energy units with the

Government's targeted level of solar energy usage being 0.4 unit . The

consumers' total budget for energy products is 5 and the unit prices of

fossil fuel and solar energy are 1 and 10, respectively. Thus, solar

energy is assumed to cost ten times as much as fossil energy currently ,

costs. Without incentives, the consumer can purchase up to 5 units of

fossil energy or 0.5 unit of solar energy. This budget constraint is

shown as a straight line between the (5, 0) and (0, 0.5) points in

Figure 1. The consumer's desired place on the budget line is point 1

where his utility is maximized. Solar energy usage is only 0.25 unit

which is less than the Government's targeted usage of 0.4 unit. In­

centives are needed to encourage the consumer to use more solar energy.

At point 2 the Government's target solar level is achieved with

either a subsidy or tax and fossil fuel use is not increased: either a

subsidy of $3.75 per unit of solar energy or a tax on fossil fuel of

-$3.00 q2 + $0.60. The tax on fossil energy is a function of the

amount of the consumers' solar energy usage. A straight tax on fossil

energy similar to the solar subsidy decreases fossil energy usage, but

it does not guarantee more solar usage. For the standard utility

function, the share of income for purchasing solar or fossil energy

remains the same; so, when the price of fossil fuel is raised through

direct taxes, the amount of fossil energy consumed is reduced with no

effect on solar energy consumption. This is why, at point 2, the solar

subsidy increased the use of solar energy with no effect on the use of

fossil energy; but if either a tax or subsidy is used to move the

consumer to point 2, he is better off than he was at point 1. Thus, the

individual would benefit from the incentive program because the

Government must increase the consumers' energy budget to achieve the

targeted solar energy level.

The targeted solar level is achieved at point 3, but the use of

fossil fuel has increased. The necessary tax is -$1.875 q2 per unit of

fossil fuel. The budget line, including this incentive, has end points

equal to the initial budget line. The consumer is better off at point

3 than at point 1 because he can now purchase more solar energy and

more fossil fuel. With this subsidy the consumer is not only paid to

increase his use of solar energy but also to increase his use of fossil

energy.

Point 4 is located on the budget line, but since it has less

utility than point 1, it was not initially chosen. rhis point would

be chosen if the tax were -$7.?0 q2 + $3.00. The net government cost

at this point is zero (i.e., taxes equal subsidies) because the point

is on the original buqget curve. This point can also be attained by

rationing fossil fuel to one unit or by regulating solar energy usage

to 0.4 unit.

With the proper mix of subsidies and taxes, any point in the

ql - q2 plane can be made desirable to the consumer, providing that ql

is less than I/Pl' A major problem with using the standard utility

function for solar energy analysis is that both types of energy must

b~ used or the utility is zero. In most processes where solar energy

is a feasible substitute for fossil fuel, the total amount of energy is

more important than having some of each type. This suggests that a

plausible objective for the consumer may be obtaining the most energy

possible within the budget constraint.

Stackelberg Example 2

The secon9 example shows how a consumer reacts if he desires to

maximize the amount of energy used. This is also equivalent to minimi­

zing the per unit cost of energy. The Government's goal (Eq. (5)) and

the consumer's budget coqstraint (Eq. (7)) are the same, but the

consumer'S utility is now:

(8)

The utility curves are straight lines denoting equal energy usage.

The value of a unit of energy is the same to the consumer, no matter

where it comes from. Subsidies and taxes are again investigated using

this new utility function.

Point 1 on Figure 2 is the desired energy mix without subsidies

or taxes. Since utility is defined as the total amount of energy,

the optima+ mix is to buy on the cheapest; and in this case, fossil

fuel. Thus, solar is not initially being used, so subsidies and taxes

must be offered before solar penetrates the market.

9

I-' o

~ ... :>t 0 ~8 til. z .... til

~

<~ ~o 0 00

tL..o 0", 00 0 E--,-Z ::>~ II 0

or

~~ 0

, , , , , ,

\ , , , ,

, , , , , , , ,

, • , • , • , •

t=O

t=-9q.+3.6

0.0 1.0

Figure 2.

, ,

, , , , ,

q1

, , , • , , , ,

, , , \

• , , • , , , . , . , , , , , , , ,

, , , •

, . , , . . . , , , , " . , , , ,

t=-2.13Q. +.m . , .

2.0 3.0 4.0 UNITS OF FOSSIL FUEL

3 t=-l.99q.

, , , , . , . . . , . . . , , . , '1

,

5.0

Impa'ct of Incentives on the Consumer I s Energy Use Assuming Total Enersy as the Utility Function

6.0

The desired energy mix at point 2 requires a tax of -$2.13 q2 + $0.07. The total energy usage has not increase d (i.e., point 1 and

point 2 are on the same utility curve), but 0.4 unit of fossil fuel ha s

been displaced with solar energy. Since the consumer utility has not

been reduced, the Government must pay for 90 percent of the solar

energy used.

At point 3, the higher utility means greater energy usage. The

tax of -$1.99 q2 is equivalent to a government payment of 98 percent

of the solar energy costs. The boundary points on the budget line are

the same as the no-tax case, so the fossil fuel user is not penalized

if he decides against participation in the solar incentive program.

Point 4 is on the original budget line so the net cost to the

Government is zero, but the utility (i.e., energy use) has been

severely reduced. This point can also be achieved by regulating or

rationing.

For a solar subsidy to impact the energy mix, it must be set to

at least 90 percent of the solar costs because of the ten-to-one price

ratio between solar and fossil energy. Unless the 90 percent subsidy

is limited, large-demand impulses would destabilize the solar sector of

our economy. The tax scheme outlined above stimulates interest in

partial displacement of fossil energy with solar and provides better

government control over solar energy demand.

Conclusion

The initial theoretical development of leader-follower games

involving solar incentives has been presented. The game theory approach

forces policymakers to evaluate potential incentives against defined

objectives. From this theoretical work, Example 2 indicates that

incentives in the form of a tax, rather than straight solar subsidies,

may provide for more control of solar demand by the Government. The

static results from this analysis can be expanded to develop proper

incentives for various points on the solar industry learning curve.

The evaluation of hoW incentives affect multiple consumers a nd/or

producers is a straightforward extension of the present analysis.

11

12

REFERENCES

1. H. von Stackelberg, The Theory of the Market Economy, (1952), Oxford University Press, Oxford, England.

2. J. Case, "Toward a Theory of Many Player Differential Games," SIAM J. on Control, 7, (1969), pp 179-97.

3. M. Simaan and J. B. Cruz, Jr., "On the Stackelberg Strategy in Nonzero-Sum Games," JOTA, 11, (1973), pp 533-55.

4. M. Simaan and J. B. Cruz, Jr., "Additional Aspects of the Stackel­berg Strategy in Nonzero-Sum Games," JOTA, 11 (1973), pp 613-26.

5. J. Green and J. Laffont, Incentives in Public Decision Making, (1980), North-Holland Pub. Co., New York, NY.

6. M. A. Salmon and J. B. Cruz, "An Incentive Model of Duopoly with Government Coordination," Automatica, V. 17, Nov. 1981 (to appear).

Distribution:

Solar Information Center 1536 Cole Blvd. Golden, CO 80401 Attn: R. Ortiz

Solar Energy Research Inst. (3) 1536 Cole Blvd. Golden, CO 80401

Jet Propulsion Laboratory (3) 4800 Oak Grove Drive . Pasadena, CA 91103 Attn: J, Becker

J. Lucas K. Terasawa

Office of Technology Asqessment U. S. Congress Washington, DC 20510 Attn: R. Rowberg

U. S. Dept. of Energy (3) Albuquerque Operations Office P. O. Box 5400 Albuquerque, NM 87185 Attn: G. N. Pappas

C. B. Quinn J. Weisiger

U. S. Dept. of Energy (5) Division of Energy Storage Systems Washington, DC 20545 Attn: J. Gahimer

U. S. Dept. of Energy (8) Division of Solar Thermal Energy

Systems Washington, DC 20585 Attn: W. W. Auer

G. W. Braun J. E. Greyerbiehl M. U. Gutstein L. Melamed J. E. Rannels · F. Wilkin::;; J. Dollard J. Easterling

U. S. Dept. of Energy (2) San Francisqo Operations Office 1333 Broadway, Wells Fargo Bldg. Oakland, CA 94612 Attn: R. W. Hughey

4000 4700 4710 4713 4714 4715 4716 4717 4720 4721 4723 4724 4726 4750 5510 5513 5520 5523 8326 8430 843], 3141 3154-3

3151

6011 4723

A. Narath J. H. Scott G. E. Brandvold J. v. Otts R. P. Stromberg R. H. Braasch J. F. Banas J. A. Leonard D. G. Schueler W. P. Schimmel D. G. Schueler (Act g ) E. C. Boes E. L. Burgess v. L. Dugan D. B. Hayes D. W. Larson T. B. Lane R. C. Reuter c. T. Yokomizo R. C. Wayne P. J. Eicker L. J. Erickson (5 )

C. Dalin (25) for DOE/TIC ' W. L. Garner (3) (Unlimited Release for

DOE/TIC) Patents J. K. Sharp

13